[FOM] Independence without forcing

[Harvey Friedman]

>
>In the archives, I found a post by Gaifman where he asks about
>V=L and "hard core independence proofs". If I interpret things
>correctly, he and I are asking nearly the same thing.  Here's a more
>precise formulation.
>
>Consider the following position:
>
>"All statements of 'real mathematics' are decided in the axiom system
>ZF + V=L + there are no inaccessible cardinals; all other statements
>are metamathematical."
>
>I am giving 'real mathematics' a loose interpretation --- almost
>anything except a Godel sentence counts. The position as stated does
>not claim that the quoted axioms are true, only that everything of
>interest is decided by them.
>
>Is the position a valid one or not? Assuming it is not, is a
>refutation in the realm of 'science-fiction', or will I discover the
>answer sometime in the coming week while I'm reading more about
>Boolean relation theory?
>

The main statement in Boolean Relation Theory is a counterexample to 
that position. The main statement is proved using Mahlo cardinals of 
finite order in

H. Friedman, Equational Boolean Relation Theory, 
http://www.mathpreprints.com/math/Preprint/show/

The second part of that paper, containing a proof that the statement 
implies Con(ZFC) and much more, is being written now, and will slow 
me down for a while on the FOM.

Let A be this main statement. The result is that

ACA proves that A equivalent to the 1-consistency of ZFC + {there 
exists an n-Mahlo cardinal}_n.

ACA is the tiny fragment of ZFC called arithmetic comprehension axiom 
scheme. This 1-consistency statement phi is

1) not provable in ZF + V = L + there are no inaccessible cardinals, 
assuming ZF is consistent; and

2) not refutable in ZF + V = L + there are no inaccessible cardinals, 
assuming ZFC + (forall n)(there exists an n-Mahlo cardinal) is 
consistent, or even just that ZFC + phi is consistent.

One does not have to use the main statement of BRT. One can also use 
basic statements arising out of the main BRT classification of 6561 
statements, or the classification itself.

If I recall, I gave some examples to what Gaifman was asking for 
which were discussed by Gaifman in later FOM postings by Gaifman, and 
also Gaifman mentioned some related work of Franzen.

One might ask if BRT can be FINE TUNED to yields a statement S with 
these properties:

3) PA proves Con(ZF) iff Con(ZFC + S) iff Con(ZFC + notS).

With some additional control of the relevant combinatorics, I believe 
that natural statements from BRT arise, S, such that

4) ACA proves S iff ZF(C) is 1-consistent.

This is close to 3), but not quite 3). To get 3), one would need to 
get, for example, a BRT statement S such that

5) ACA proves S iff "the least n such that ZFC_n is inconsistent, if 
any, is even".

Here the n refers to the number of quantifiers allowed in the 
Replacement axiom, say with full separation allowed.

This can probably be done in BRT, with strong control over the 
combinatorics involved, although one has to use a fair amount of 
finesse in order to keep this appropriately natural.

I don't regard such FINE TUNING as of a high priority for BRT right 
now. The highest priority is in classifications and strengthening 
connections with general mathematical culture. The issue of extending 
the scope of BRT to higher cardinals than Mahlo cardinals is also of 
higher priority than reducing down to ZFC.

Harvey Friedman